Region of controllability for optimal control

Question

I have an optimal control question which yields:

$x=A e^{4t}\begin{pmatrix}1\\-1\end{pmatrix}+B e^{2t}\begin{pmatrix}1\\1\end{pmatrix}+\begin{pmatrix}1\\3\end{pmatrix}u^*$

For $u^* = \pm1$

So we can see from the eigenvalues we have these two repulsive nodes centred at $(1,3)$ and $(-1,-3)$

Is the region of controllability just the rectangle created from the lines of the eigenvectors?